EITN90 Radar and Remote Sensing Lecture 5: Target Reflectivity

Transcription

1 EITN90 Radar and Remote Sensing Lecture 5: Target Reflectivity Daniel Sjöberg Department of Electrical and Information Technology Spring 2018

2 Outline 1 Basic reflection physics 2 Radar cross section definition 3 Scattering regimes 4 High-frequency scattering 5 Examples 6 Conclusions 2 / 53

3 Learning outcomes of this lecture In this lecture we will Study the properties of electromagnetic waves Define the radar cross section (RCS) Understand basic scattering and reflectivity physics Understand how two or more scattering centers interfere Illustrate some high-frequency scattering effects 3 / 53

4 Outline 1 Basic reflection physics 2 Radar cross section definition 3 Scattering regimes 4 High-frequency scattering 5 Examples 6 Conclusions 4 / 53

5 Electromagnetic waves The time-harmonic Maxwell s equations in linear media are (where all fields have time dependence E(x, y, z, t) = E(x, y, z)e jωt ) { E = jωb = jωµh H = jωd = jωɛe where ɛ and µ are the permittivity and permeability of the material. A plane wave propagating in direction k is given by (where we use e jk R = jke jk R, or jk) { E = E0 e j(ωt k R) H = H 0 e j(ωt k R) { k E0 = ωµh 0 k H 0 = ωɛe 0 The cross products means (E 0, H 0, k) is a right-hand triple like (ˆx, ŷ, ẑ). In addition, the equations imply k = ω ɛµ and the ratio E 0 / H 0 = µ/ɛ = η, where η is the wave impedance. 5 / 53

6 Vector analysis, linear algebra The vectors have three components, one for each spatial direction: E = E x ˆx + E y ŷ + E z ẑ In particular, the position vector is r = xˆx + yŷ + zẑ. Vector addition, scalar product and vector product are r 1 r 2 r 1 r 2 r 2 r 2 ϕ r 1 ϕ r 1 + r 2 r 2 cos ϕ r 1 Addition: r 1 + r 2 = (x 1 + x 2 )ˆx + (y 1 + y 2 )ŷ + (z 1 + z 2 )ẑ. Scalar product: r 1 r 2 = r 1 r 2 cos ϕ = x 1 x 2 + y 1 y 2 + z 1 z 2. Vector product: orthogonal to both vectors, with length r 1 r 2 = r 1 r 2 sin ϕ, and r 1 r 2 = r 2 r 1. ˆx ŷ = ẑ, ŷ ẑ = ˆx, ẑ ˆx = ŷ 6 / 53

7 Vector analysis, differentiation (not necessary to understand the book) The nabla operator is = ˆx x + ŷ y + ẑ z The gradient, divergence, and curl operations are g = g g ˆx + x y ŷ + g z ẑ E = x E x + y E y + z E z E = x ˆx E + y ŷ E + z ẑ E Cartesian representation (useful in numerics): [E] = [ E] = x E x E y 0 } 1 0 {{ E z } =[ˆx E] + y E x E y 1 } 0 0 {{ E z } =[ŷ E] ) ( Ex E y and E z z E x E y E z }{{} =[ẑ E] 7 / 53

8 Misprinted equations Unfortunately, the vector cross product sign,, is sometimes not printed in the book. The correct versions of the affected equations are below: (6.5) P = 1 2 Re(E H ) W/m 2 (6.6) E total ˆn = (E inc + E scat ) ˆn = 0 Page 216, paragraph 2, line 3: ˆn E scat = ˆn E inc Comment: the vector product has amplitude ˆn E = ˆn E sin ϕ = E sin ϕ where ϕ is the angle between the unit vector ˆn and the vector E. Hence, ˆn E represents the part of E orthogonal to ˆn (tangential to the surface if ˆn is a surface normal). 8 / 53

9 Frequency and wavelength x λ H z y E The electric field is E(x, y, z, t) = E 0 cos(ωt kz)ˆx. The wavelength λ is the periodicity in z, determined by (using ω = 2πf and ωt kz = ω(t kz/ω) = ω(t z/v)) λf = v k = ω v = 2π λ The polarization corresponds to the direction of the electric field. The wave depicted above is linearly polarized in the x-direction. 9 / 53

10 Polarization Vertical Horizontal Right hand Left hand See the animation program EMANIM. 10 / 53

11 Right-hand rules Let the propagation direction k be along the thumb. At any time, E, H, and k are orthogonal to each other. Linear polarization Circular polarization V H rotation direction k k E oscillating along Horizontal or Vertical direction, H along the other. E rotating along Right or Left hand fingers, H rotating the same but at right angle. 11 / 53

12 IEEE definition of left and right With your right hand thumb in the propagation direction and fingers in rotation direction: right hand circular. E(t) y x right-polarized forward moving z E(t) y x left-polarized forward moving z E(t) z y x left-polarized backward moving E(t) z y x right-polarized backward moving 12 / 53

13 Refractive index and wave impedance From the expression of a plane wave E = E 0 e j(ωt k R) = E 0 e jω(t k ω k k R) where k/k = ˆk is the propagation direction, it is seen that only the projection of the position vector R in the propagation direction ˆk, ˆk R, matters, and the speed of propagation is v = ω k = 1 ɛµ = c n, where n = ɛµ ɛ 0 µ 0 The refractive index n is the speed of an electromagnetic wave in a material (v), relative to the speed in vacuum (c). The wave impedance in the material is given by (denoted by η or Z) E µ H = ɛ = η = Z 13 / 53

14 Plane waves vs spherical waves The plane wave is a beautiful theoretical tool, but it typically only applies locally around the target or transmitter/receiver. λ transmitter D target The wavefront is spherical close to the transmitter, but approximately plane at the target if the range R satisfies R R 2D2 λ where D is the diameter of the target and λ the wavelength. 14 / 53

15 Induced currents Assume the target is made of metal. The electrons move around so as to cancel the field inside the metal, quantified through the boundary condition (zero tangential electric field) E tot ˆn = (E inc + E scat ) ˆn = 0 A more general boundary condition is E tot tan = Z S ˆn H tot, where Z S is the surface impedance (Z S = 0 for perfect conductors). This does not change the general results in this lecture. 15 / 53

16 Radiation Since the incident field is oscillating like e jωt, the induced surface currents J S and surface charges ρ S on the target will oscillate with the same frequency and radiate a scattered field E scat. ( E scat (R f ) = jωµj(r s )g(r f R s ) + ρ S(R s ) ɛ ) g(r f R s ) ds(r s ) The function g(r f R s ) = e jk R f Rs 4π R f R s is called the Green s function, corresponding to radiation at field point R f from a unit point source at R s. 16 / 53

17 Scattering theory definitions At large range from the scatterer, the scattered field can be written E scat = e jkr R F (ˆk scat ) where F (ˆk scat ) is the far field amplitude in scattering direction ˆk scat. Given knowledge of scattered electric and magnetic fields E s and H s on a surface S enclosing the scatterer, this is F (ˆk) = jk 4π ˆk [ˆk (ˆn η0 H s ) + E s ˆn] e jkr ds The bistatic scattering cross section is S σ (ˆk inc, ˆk scat E scat (ˆk scat ) ) = lim R 4πR2 E inc (ˆk inc ) 2 2 F (ˆk scat ) = 4π E inc 2 2 and the monostatic radar cross section at incident direction ˆk is σ(ˆk) = σ (ˆk, ˆk) 17 / 53

18 Simulations The scattering theory can be implemented numerically. This provides excellent tools to compute and visualize the scattered field / 53

19 Outline 1 Basic reflection physics 2 Radar cross section definition 3 Scattering regimes 4 High-frequency scattering 5 Examples 6 Conclusions 19 / 53

20 Radar cross section In this chapter, power density is denoted by P instead of Q as in Chapter 1. The IEEE definition of RCS is σ def = lim R 4πR2 Escat 2 E inc 2 which is motivated by intercepted power σp inc and isotropically radiated scattered power density P scat = (σp inc )/(4πR 2 ). 20 / 53

21 Factors affecting the RCS The bistatic RCS of a target depends on the following factors: Target geometry and material composition. Direction of transmitter relative to target. Direction of receiver relative to target. Frequency or wavelength. Transmitter polarization. Receiver polarization. In a monostatic setting, transmitter and receiver are co-located. 21 / 53

22 Polarization scattering matrix It is sometimes necessary to keep track of polarization. In general, the polarization scattering matrix (PSM) S is defined by E scat (ˆk scat ) = S(ˆk scat, ˆk inc ) E inc (ˆk inc ) where the arguments ˆk scat and ˆk inc are often left out to write E scat = S E inc. The PSM can be represented in linear polarization ( ) E scat V EH scat = ( SVV S VH S HV S HH ) ( E inc V E inc H or circular polarization ( ) ( ) ( ) E scat R SRR S EL scat = RL E inc R S LH S LL EL inc ) 22 / 53

23 Converting between polarizations The relation between LP and CP in transmission is ( ) E t R = 1 ( ) ( ) ( ) 1 j E t H E t H = 1 ( ) ( ) 1 1 E t R 2 1 j 2 j j E t L E t V and in reflection we have (due to different propagation direction) ( ) E r R = 1 ( ) ( ) 1 j E r H 2 1 j E r L E r V E t V E t L ( ) E r H = 1 ( ) ( ) 1 1 E r R 2 j j Hence, the polarization scattering matrix in LP and CP are related by ( ) SRR S RL = 1 2 S LR S LL ( 1 j 1 j E r V ) ( SHH ) ( S HV 1 ) 1 j j S VH S VV E r L Note that the sense of rotation of circular polarization changes with each reflection. 23 / 53

24 Outline 1 Basic reflection physics 2 Radar cross section definition 3 Scattering regimes 4 High-frequency scattering 5 Examples 6 Conclusions 24 / 53

25 Sphere scattering In the following slides, scattering from an oblong object is shown. Note the outmost spherical shell corresponds to a material layer absorbing outgoing radiation, and does not correspond to a physical region. 25 / 53

26 Low frequency: Rayleigh scattering f = f 0 /8 f = f 0 /4 f = f 0 /2 Incident field Scattered field Total field 26 / 53

27 Intermediate frequency: resonant scattering f = f0 f = 2f0 f = 4f0 Incident field Scattered field Total field 27 / 53

28 High frequency: optical scattering f = 8f0 f = 16f0 f = 32f0 Incident field Scattered field Total field 28 / 53

29 Scattering mechanisms 29 / 53

30 Outline 1 Basic reflection physics 2 Radar cross section definition 3 Scattering regimes 4 High-frequency scattering 5 Examples 6 Conclusions 30 / 53

31 Superposition from several scatterers E 0 e jk R ˆk R 1 R 1 ˆk R 2 R 2 0 ˆk R 3 R 3 When several scatterers are subjected to an incident wave E 0 e jk R, the backscattering is (complex addition) N σ tot = σi e j2k R i i= / 53

32 Phasor addition Two scatterers spaced by 2λ. Strong interference when scatterers have equal amplitude, dominated by the strong scatterer when they are very different. 32 / 53

33 Specular scattering When the surface normal ˆn of a relatively flat surface points toward the radar, there is little variation of ˆk R over the surface. Hence, the phase does not change much, and we have coherent addition: σ specular = 4π A2 cp λ 2, A cp = area of constant phase Rc λ L cp 2 σ = πr c1 R c2 length of constant phase, R c = radius of curvature double curved surfaces σ cyl = 2π λ R cl 2 cylindrical surface 33 / 53

34 End-region scattering Scattering from a metal plate shows significant side-lobes. At off-specular directions, only the edges are scattering in the back-direction. The effect is due to truncation of currents on the flat plate. 34 / 53

35 Metal plate at different orientations Scattering from edges is stronger than from corners. 35 / 53

36 Edge diffraction When a wave is incident on an edge, a line source current is induced. At oblique incidence it radiates in a cone. Monostatic return only at normal incidence, σ L 2 /π. 36 / 53

37 Multiple bounces When two specular reflections combine at 90 angle, strong backscattering occurs, so called corner reflectors. 37 / 53

38 Corner reflections Incident field Scattered field Total field 38 / 53

39 Multiple bounces The specular reflections do not need to be at flat surfaces. Many different combinations can occur. When designing stealthy objects, it is important to find shapes with as little corner reflections as possible. 39 / 53

40 Outline 1 Basic reflection physics 2 Radar cross section definition 3 Scattering regimes 4 High-frequency scattering 5 Examples 6 Conclusions 40 / 53

41 Metal plate, different methods Diffraction important at low levels of scattering. 41 / 53

42 A7 aircraft 42 / 53

43 A7 aircraft 43 / 53

44 A7 aircraft 44 / 53

45 Stovepipe aircraft 45 / 53

46 Stovepipe aircraft 46 / 53

47 Stovepipe aircraft 47 / 53

48 Stovepipe aircraft 48 / 53

49 Stovepipe aircraft 49 / 53

50 Stovepipe aircraft 50 / 53

51 Stovepipe aircraft 51 / 53

52 Outline 1 Basic reflection physics 2 Radar cross section definition 3 Scattering regimes 4 High-frequency scattering 5 Examples 6 Conclusions 52 / 53

53 Conclusions We have reviewed basic scattering theory and how it relates to RCS. Three different scattering regimes: Rayleigh, resonance, optical. Interaction between multiple targets. Scattering mechanisms: dipole, surface waves, specular, multiple bounces, end regions, edge diffraction, discontinuities. 53 / 53